Error estimates for the Runge-Kutta discontinuous Galerkin method for the transport equation with discontinuous initial data

Bernardo Cockburn, Johnny Guzmán

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18 Scopus citations

Abstract

We study the approximation of nonsmooth solutions of the transport equation in one space dimension by approximations given by a Runge-Kutta discontinuous Galerkin method of order two. We take an initial datum, which has compact support and is smooth except at a discontinuity, and show that, if the ratio of the time step size to the grid size is less than 1/3, the error at the time T in the L2(ℝ RT)-norm is the optimal order two when DIT is a region of size O(T1/2 h1/2 log 1/h) to the right of the discontinuity and of size O(T1/3 h2/3 log 1/h) to the left. Numerical experiments validating these results are presented.

Original languageEnglish (US)
Pages (from-to)1364-1398
Number of pages35
JournalSIAM Journal on Numerical Analysis
Volume46
Issue number3
DOIs
StatePublished - Nov 10 2008

Keywords

  • Discontinuous Galerkin methods
  • Error estimates
  • Hyperbolic problems

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